Proposition 3 (Reduction to isotonic regression [1]}, [2]}, [3]})  Let \(\sigma \in \mathfrak {S}_n\)  that sorts \({z}\)  decreasingly, i.e. \(z_{\sigma (1)} \ge \ldots \ge z_{\sigma (n)}.\) 
Let \({x}\)  be a solution to isotonic regression on \({z_\sigma }- {\tilde{w}}\) , i.e.
\({x}= _{x^{\prime }_1 \le \ldots \le x^{\prime }_n} \frac{1}{2}\Vert {x}^{\prime } - ({z_\sigma }- {\tilde{w}})\Vert ^2\) 
